Download Adiabatic processes in the ionization of highly excited hydrogen atoms

Atoms, Molecules
zo.=.,
far Physik D and Clusters
Z. Phys.D- Atoms,Moleculesand Clusters 11, 1-14 (1989)
© Springer-Verlag 1989
Adiabatic processes in the ionization
of highly excited hydrogen atoms
H.P. Breuer and M. Holthaus
PhysikatischesInstitut,Universit/itBonn,Nussallee12, D-5300 Bonn 1, FederalRepublicof Germany
Received25 August1988;finalversion 19 September1988
Adiabatic and non-adiabatic processes in the dynamics of periodically driven quantum
systems are studied employing the Floquet picture of quantum mechanics. The validity
of N-niveau approximations in the Floquet theory is investigated. A method is described
which allows the separation of fast (periodic) and slow (parametric) time-dependence
and which yields a transparent description of the dynamical Landau-Zener mechanism.
Using the model of surface-state electrons it is demonstrated that adiabaticity is decisive
for the description of ionization experiments on highly excited hydrogen atoms; furthermore, an ionization mechanism based on the sudden onset of Landau-Zener transitions
is proposed and shown to yield values for the ionization threshold which are in good
agreement with experimental data. The connection to classical mechanics is considered
by establishing the semiclassical limit of Floquet dynamics.
PACS: 03.65; 05.45; 32.80
1. Introduction
Although the interaction of atoms and molecules with
time-periodic external fields is a very old subject in
quantum mechanics, recent experiments in this area
have discovered unexpected phenomena which still
are not fully understood theoretically. A particularly
puzzling example is given by the behaviour of highly
excited hydrogen atoms in strong microwave fields
which has been investigated experimentally by Bayfield and Koch [1, 2]. In these experiments, one observes ionization although the ionization energy of
the initial stationary state corresponds to the energy
of up to several hundred microwave photons [2]. Furthermore, the ionization rate depends sensitively on
the strength but hardly on the frequency of the microwave field. These results have stimulated many theoretical activities. Since the full solution of the timedependent Schr6dinger equation in 3 space dimensions is beyond present-day abilities one has to rely
on simple models. One of the most important models
in this context is the model of surface-state electrons
(SSE model) in the presence of an oscillating field
which has been investigated within the framework of
both classical [3, 4] and quantum [5-7] mechanics.
Whereas even the first classical numerical simulations
[3] took into account the fringe fields of the microwave cavity which in the rest frame of the atom appear as a slow rise and decrease of the external field,
the quantum calculations usually rest on the assumption that the effect of the turn-on of the external field
can be neglected [7]. As we shall show in this paper,
this assumption is, in general, invalid; on the contrary,
adiabatic behaviour, which in turn results from the
slow turn-on of the field, plays a major r61e in the
dynamics of the solutions of Schr6dinger's equation.
Following the general reasoning of prior investigations [8, 9] where it has been shown that for Hamiltonians periodic in time adiabaticity is intimately connected to the Floquet picture of quantum mechanics,
we use the SSE model for a detailed study of those
adiabatic processes that govern the response of highly
excited hydrogen atoms to the entrance into a microwave cavity. To understand the initial step of the ionization mechanism we restrict ourselves to a discussion
of the dynamics in the bound state part of the underlying Hilbert space.
As a main result we show that the Floquet picture
is very well suited for describing situations where the
amplitude of the external field is not constant in time
since in this picture adiabatic behaviour expresses itself as conservation of the occupation probabilities
of the Floquet states when the driving amplitude is
varied. For the particular case of the SSE model, we
show that Landau-Zener type transitions among Floquet states due to the appearance of multiple avoided
crossings of quasienergies above certain critical values
of the field strength provide a transparent mechanism
for exciting higher Floquet states; these critical values
are related to the experimentally observed ionization
threshold.
The material of our paper is organized as follows:
Chap. 2 contains the basic elements of Floquet theory
and mainly serves to establish our notation. In
Chap. 3 we show that numerical calculations of Floquet spectra within a finite basis set are subject to
certain errors which have to be carefully controlled;
these results are applied to obtain stable quasienergy
spectra for the SSE model. In Chap. 4, which has to
be regarded as the central one of this paper, we present a general discussion of adiabaticity in the Floquet
picture; again we use the SSE model to support our
theoretical conclusions. Finally we sum up our findings in Chap. 5 and point out how to apply semiclassical quantisation rules to calculate quasienergy spectra. This may be of interest since the classical counterparts of driven quantum systems exhibit, in general,
chaotic behaviour.
imposing (2.4) as a boundary condition. Formally,
the operator
:=H(x, t)--ia t
(2.6)
acts on a Hilbert space of time-periodic functions [11]
with scalar product
1 r
((u] v>>.'=~ oS dtSdx u*(x, t)v(x, t);
the completeness relation in this space reads
u=(x, t) u* (x', t') = a (x -- x')
Tar (t - t'),
u=(x,t)=u~(x,t)e imp',
meZ,
Let us consider a Hamiltonian with periodic time dependence
(2.1)
where T is the period and x denotes all spatial and
spin degrees of freedom. According to Floquet's theorem 1-10] there are solutions of the Schrrdinger equation
t)- i8t) O(x, t) = 0
(2.2)
corresponding to the eigenvalues
e~=e~+mco,
meZ.
(2.10)
Hence e can be written as an array
(2.3)
where u is periodic in time
u(x, t)=u(x, t + T);
(2.4)
e is referred to as Floquet index or quasienergy.
In order to obtain such solutions one has to solve
the eigenvalue equation
(H(x, t)-i~?t)u= (x, t)= G u=(x, t)
(2.11)
ui (x, t) u* (x', t) = c5(x -- x').
(2.5)
(2.12)
J
It easily follows from (2.3) that the solution 0(x, t)
of Schr6dinger's equation is independent of m. Thus
the whole class (2.9) corresponds to one physical state
As an instructive example we consider the forced
harmonic oscillator in (1 + 1) space - time dimensions
given by the Hamiltonian
H(x,t)=~---fi+@x2+2xsincot.
that can be written in the form
(x, t)= u (x, t ) e - i "
(2.9)
and the functions u i(x, t) form a complete system at
each instant of time:
2. Elements of Floquet theory and notation
(H(x,
(2.8)
where a r denotes the T-periodic a-function.
If ui(x, t) is a solution of (2.5) with eigenvalue
ej we immediately obtain a whole class of solutions
~=(j, m)
H(x, t)= H(x, t + T),
(2.7)
(2.13)
From the well-known solutions of Schradinger's
equation
t
Ill(X , t) =)~j(X -- J(t)) e-iE't ~oL(z) dz +i#y(x -y)
(2.14)
(gj is the oscillator eigenfunction with energy Ej =
O)o(J+½), L is the classical Lagrangian evaluated
along the classical periodic trajectory y(t)), we obtain
the eigensolutions u~(x, t) by isolating the time-peri-
odic part:
= Zj(x-- y(t)) e' o~Ld~-~ ! Ldz+,}'(x--y)
= uj (x, t) e- i~jt.
"~
\
~o
/
(2.15)
Hence the Floquet spectrum of the forced harmonic
oscillator is given by
3. N-niveau approximations in Floquet theory
e ~ , = E j - l ~ Ld'~ +mco
22
1
+-me)
= E j + 4#0) 2 1--/co°\2
(2.16)
where m is any integer and j = 0, 1, 2, 3, ....
From the point of view of physical applications
one encounters Hamiltonians of the form
H (x, t) = H o + 2 x sin cot
In this chapter we are concerned with the numerical
solution of the eigenvalue equation (2.5) where H(x, t)
is of the form of (2.17). A commonly adopted strategy
consists in transforming the Schr6dinger equation
(2.2) into a matrix equation
iOt~,(t)=E,t~,(t)+ 2sincot ~ D.m~.,(t)
using an eigenbasis of Ho
Ho ~o, = E, ~o,,
----;
x>0
x
+oe;
(3.2)
such that
D,,, = (q%l x lq)m)
(3.3)
is the matrix of the dipole operator x. To solve (3.1)
on a computer one has to truncate this system to
a finite dimensional one, since only a finite basis set
{~Ol, ~0z, ..., q~N}
(3.4)
can be handled numerically. Within this N-niveau approximation one calculates the monodromy operator,
i.e. the time-developement operator U(m(T, 0) for a
complete cycle. From the Floquet theory it follows
that diagonalisation of U(m(T, 0) yields the eigenvalues
e
V(x) =
n = 1, 2, 3.....
(2.18)
V(x)
1
(3.1)
(2.17)
where Ho is the time-independent Hamiltonian of an
atomic or molecular system and the second term describes the interaction with a single mode radiation
field in the dipole approximation; 2 denotes the field
strength and co its frequency. In most cases the spectrum of Ho contains a continuous part and hence
the Floquet spectrum of (2.17) covers the whole real
axis, i.e. in a strict mathematical sense there are no
normalisable Floquet states [12]. The situation can
be seen in close analogy to the static (DC) Stark effect:
bound states of Ho turn into resonances described
by complex Floquet indices with a negative imaginary
part.
The model we shall mainly use to support our
general conclusions is the model of surface-state electrons (SSE model) given by a (1 + 0-dimensional Hamiltonian of the form (2.17) where
H o = T1p ~z +
place in the bound space and therefore it should be
sufficient to decouple the continuum and to consider
the 'bound space projected' dynamics [7]. This is a
reasonable assumption as long as the time scale of
decay set by the imaginary part of the Floquet indices
is long compared to the time scales of these dynamical
processes.
x_<0.
(2.19)
This model had originally been devised in order to
describe electrons bound by their image charge to
a surface of liquid helium and has meanwhile frequently been used to investigate the microwave ionization of hydrogen atoms in both classical [-3, 4] and
quantum [5-7] calculations; the reduction to one
space dimension has been justified in [6]. Although
ionization ultimately involves the coupling to the continuum it has been argued that the initial step of the
ionization process must be due to some process taking
J
)j=l
. . . . . N~
where the 8~N) are the Floquet indices (determined
rood co) in the N-niveau scheme.
For our investigation of adiabaticity in Chap. 4
we do not only need the Floquet indices for a specific
value of the field strength 2 but rather we have to
know the whole 2-dependence
e~=~(2).
(3.5)
It is the purpose of the following considerations
to show that the N-niveau approximation will lead
to systematic errors of the functions
e~N)= e~m(2)
(3.6)
e/w
,
~k'to
U
r
3w
2w
1to
I
-lw
)
--" E I
El E2 "'"
-2to
a
3t~
2~
]to
-laJ
-2w-~~o
i
2
3
A/~ 2
Fig. 1. Quasienergies for the driven oscillator (2.13) calculated within
an ( N = 10)-niveau approximation and organized in two Brillouin
zones, Parameters chosen are: c9=1.0 g, o)o=0,9 1x. Indices e are
chosen such that c~= (j, 0) denotes the Floquet index continuously
connected to Ej=(,Oo(Jq-½)
for every finite N which have to be carefully controlled.
A first indication for these errors can be deduced
from the trace identity well known from the theory
of periodic differential equations [13]
N
S dttr(Htm(t))
e}m(2) =
j=l
(mode)).
(3.7)
O
The right hand side of this equation is 2-independent
for pure harmonic time-dependence and equal to the
sum of the first N energy eigenvalues (mod co) of the
static hamiltonian, i.e. we have
N
N
e}m(2) = Z Ej
j=1
(mode))
(3.8)
j=l
for all 2. This constraint is, however, N-dependent
and hence unphysical.
As an example we consider the driven oscillator
(2.13) within an (N=10)-niveau approximation, i.e.
we solve (2.5) numerically in the space spanned by
the 10 lowest lying oscillator eigenfunctions. The resulting Floquet indices e~1°)(2) are displayed in Fig. 1
which has to be contrasted to the exact solution given
by (2.16). According to this formula all the exact indi-
:
El' E2 "'"
i
EN-j
SN
b
Fig. 2, a Flow of the center X ~ of the distribution (3.10) in an
infinite dimensional Hilbert space, b as a, but in a truncated Hilbert
space
ces e~(2) show the same quadratic 2-dependence, a
fact which is incompatible with (3.8) for any finite
N.
The relevance of (3.8) can be clearly seen from
the figure: Whereas for small 2 the quasienergies corresponding to the lower lying states are approximately correct, i.e. quadratically increasing for the parameters chosen, the other indices must decrease in order
to satisfy (3.8). Obviously the 'correct' and the
'wrong' levels influence each other and produce a
pronounced avoided crossing structure such that, for
higher field strength, Fig. 1 has nothing to do with
the exact solution in the infinite Hitbert space.
This potentially dangerous effect of basis truncation becomes more transparent from the following
argument: Decomposing
u,(x, t) = ~ a}~,~q~j(x) e-ikwt
(3.9)
j,k
a Floquet state u~ can be characterised by a normalised distribution
(Ej, kco) ~ Ia ~ l 2.
(3.10)
Projected onto the E3--ko9 plane (see Fig. 2a), this
mapping can be visualised as a distribution with
center
e/w
X'~)(2)-( Sj'kEjla}~')k[;~-{((u~[II°lu~))~
--~Xj, k k o ]a}~,)
k 12]=\ ((u~[ it~t ]U~ ]"
(3.11)
Obviously, for 2 = 0 the eigensolutions are given by
u~(x, t)tx=o = q~o e-ik°~¢,
c~= (jo, ko),
(3.12)
with a delta-function like distribution and center
X{~)(2 = 0) = (kE0J;).
(3.13)
.....
(72)0.0
With increasing Z the corresponding distribution obtains a finite width (schematically depicted by patches
in Fig. 2) and its center moves in the above constructed plane.
By repeated use of the Hellman-Feynman theorem
we derive the following equation for the 'parametervelocity' (for all hamiltonians of the general form
{sT)_
(6o)~
(6~)~
(66)-(7~)_
(~9)--~
(2.17)):
(65)~.
-0.5
5
10
V/cm
This equation, of course, is valid for both the exact
solution and the N-niveau-approximation. Whereas,
however, in the infinite Hilbert space the 2-flow given
by (3.14) is unhindered (see Fig. 2a), in the truncated
case it is restricted to a strip
SN := {(x, y) IE~ <=x <_ EN}
(3.15)
which necessarily changes the global properties of the
flow (see Fig. 2 b). If a center X~(2) reaches the truncation induced boundary of SN the velocity vector must
considerably change its direction which according to
(3.14) gives rise to a large curvature term 2# 2 s J # 2 2.
Taking this as a signal for avoided crossings of Floquet eigenvalues we conclude that truncation of the
basis set inevitably leads to the occurence o f ' w r o n g '
avoided crossing structures.
Needless to say, the mathematical cause underlying our discussion is the fact that the closed unit
sphere in the infinite Hilbert space is non-compact
whereas it is compact in the finite dimensional approximating subspaces.
We remark that our argumentation may also serve
as a starting point to investigate the appearance of
real avoided crossings, since there are physical constraints, for example the 'boundary' defined by the
ground state energy of H o.
Let us now illustrate this general discussion using
the SSE model reviewed in Chap. 2. In atomic units
b
0
a0
A
V/cm
Fig. 3. a Quasieuergies for the SSE model (cf. (3.16)) calculated in
a basis of 21 states (Nmin= 55, Nmax= 75). The frequency corresponds
to 9.92 Ghz. Levels are marked by principal q u a n t u m numbers of
Ho. b as a, but using a basis of 101 states (Nmi,=55, N,,ax= 155).
Only the levels corresponding to those in a are plotted. The arrow
marks a state used in Chap. 4 to demonstrate Landau-Zener transitions
changes its monotonic behaviour, then e73 and so
on. This is a typical indication of truncation errors
fully in agreement with the naive picture of Fig. 2b
since the corresponding centers X (vS), X (74), X (73) etc.
succesively reach the critical boundary zone of the
strip (3.15). In fact, a repetition of the calculation within a slightly larger basis set (Ninon= 55; N~ax=85) already leads to the disappearance of all the avoided
crossings of Fig. 3a. In Fig. 3b we plot the quasienergies 855-875 which result from an extended calculation (Nm~,= 55; Nm,x= 155). For small 2, all the indices
decrease monotonically; above certain values of 2 the
quasienergies e75, e74 and eva are seen to undergo
several very narrow avoided crossings with levels resulting from higher lying states. This result turns out
to be stable with respect to further increase of Nm,x.
the time-dependent Schr6dinger equation is given by
1-5-7]
iOtOn(t)
1
-
Nma×
2nZ~.(t)+2sino)t
~
D.,.¢,.(t),
(3.16)
m = Nmin
where the dipole matrix D.,. is known analytically.
It should be noted that the basis set employed in
(3.16) is obtained from a twofold truncation at n =
Nmi. and n = ~m.x"
In Fig. 3 a we show the Floquet spectrum obtained
in a relatively small basis of 21 states (Nm~.=55,
Nm~x=75). Even at very low values of the field
strength 2 the index e75 behaves different from the
rest of the spectrum, at slightly larger values 874
(98)0_~5
(94)-----
I
•
I : .......
4 .........
" .•.
I
"
I
I
.'"
1 ...... : . L . : . .
(81) /
(95)
(8o)._ (87)-...~
.... - ( 8 8 ) ~ i ; z ..... ::i .....
5" ......... '
t~)--(86)--(89) .....
(85)
::i
:: ..........
:
(90)
(9o1-~
(84) "I
(91)--
(83),
(xoo)
02)
"...,
(97)
........................ % - 1 7 :
•.j
(82)
(93)--
•'
...... 122::..:
.......... : .........
< . ...... ::~
....
i
""'..i
........ ::
• "......... :............ ,i
"
-0.5
i
.:
..... : "
3
2
V/cm
4
F i g . 4. S t a b l e q u a s i e n e r g i e s f o r t h e
S S E m o d e l (60/2 ~ = 9.92 G H z )
originating from the stationary
states with principal quantum
n u m b e r s n = 80 . . . . . 100
A
Naturally, stability (i.e. truncation independence)
is a minimal requirement on a physical model. The
lesson to be learned from the preceeding discussions
may be summarized as follows:
(i) Any numerical calculation within a finite basis set is subject to the trace identity (3.8) and yields
at least some wrong levels which have to be identified
and discarded.
(ii) Stability of the remaining levels with respect
to the basis size must be guaranteed.
(iii) Special care has to be taken in order to distinguish artificial, i.e. truncation induced, avoided crossings from possible real ones, since the latter, as we
know from previous work [8, 9], dominate non-adiabatic effects.
It is a non-trivial fact that this procedure which demands a certain amount of skill actually yields stable
results.
Our next question concerns the occurence of real
avoided crossings in the quasienergy spectra of the
SSE model. Again, an important clue can be guessed
from the oscillator formula (2.16): If co is less than
the level spacing (no, all quasienergies decrease with
increasing )~ whereas they increase for co>co 0. For
the hydrogen atom and a microwave frequency of
9.92 GHz the levels with principal quantum numbers
n = 87, 88 are approximately at resonance, for lower
levels the sparing between adjacent levels is bigger
than co. Therefore it can be expected from the oscillator formula that levels with n < 87 move ' d o w n ' which
is confirmed in Fig. 3b. On the other hand, for n > 8 8
Fig. 5. Stable quasienergies for the SSE
model (a~/2 n = 9.92 GHz). F o r testing
adiabaticity in Chap, 4 the states n = 64, 74
have been used as initial states
0
5
10
V/cm
15
20
A
the level spacing is less than the frequency and the
levels should move 'up'. Hence the 'resonance point'
is of special interest, since in its neighbourhood there
are both types of levels which is a necessary condition
for an avoided crossing. This simple argument is confirmed in Fig. 4 where the quasienergies originating
from the stationary states with quantum numbers
n = 80 to n = 100 are plotted. These indices result from
a calculation employing 101 basis functions and have
been found to be stable; in particular, the avoided
crossing structure is correct.
To give an impression of the actual complexity
of the quasienergy spectra in the experimentally relevant interval of field strength we plot in Fig. 5 the
stable result of a 101 state calculation (the upper 50
and lower 7 indices have been removed). Whereas
the 'lower' quasienergies behave regular the 'upper'
ones undergo many avoided crossings on a very narrow scale.
We remark that these calculations are far more
demanding than solving the initial value problem for
the SSE model: Even without counting diagonalisations the numerical effort to obtain Fig. 5 is equivalent to propagating an initial state over 104 cycles
within a basis of 101 functions.
4. Adiabatic and non-adiabatic behaviour
in time periodic external fields
The adiabatic theorem [14] is one of the most important tools for the investigation of the time-dependent
Schr6dinger equation in the limit of slow time variation of a Hamiltonian H(z). Considering the timeinterval 0 < z < ~ and introducing the dimensionless
time-parameter
"C
s :=--
(4.1)
To
Schr6dinger's equation reads
i 0~ 0 r. (s) = T, H (s) 0 To(s),
Oro (S) ~ e
-iT~ids'E(s')
o
~
9is),
(4.6)
where
(s) q~(s) = E (s) q~(s).
(4.7)
The eigenvectors q~(s) are supposed to be normalised
(~0(s)l q~(s)) = 1
(4.8)
and their phases have to be chosen such that [14]
@(s) ~s q)(s))=0.
(4.9)
The fact that this requirement can, in general, not
be satisfied globally in a higher dimensional parameter space leads to the well-known Berry-phases
[15, 16].
The adiabatic theorem is valid even if the spectra
of the one-parameter family H(s) are unbounded or
contain a continuous part and it can be extended
to the case of degeneracies [17]. The degree of adiabaticity, i.e. the magnitude of T~ necessary to guarantee
adiabatic behaviour, depends sensitively on details of
the spectra and the smoothness of the parameter variation [18-].
There is an extension of the adiabatic theorem
to the Floquet picture of quantum mechanics which
has been formally proven in [19]. To make our assertions more transparent we should like to proceed
along different lines.
We start from the fact that strict periodic timedependence is not what is encountered in real experiments. For example, Rydberg atoms entering a microwave cavity see a slowly rising field which has to
be described by a time-dependent field strength
2 = 2(~-).
z
(4.10)
(4.2)
N o w the Schr6dinger equation
where
4,To(s) := ~,(s %)
(4.3)
and
0_< s -< 1.
(4.4)
Assuming the system to be initially in a non-degenerate eigenstate of H(0)
Or.(0)---- go(0),
the adiabatic theorem states that the solution of (4.2)
converges in the limit T~ ~ oo to an eigenstate of H(s)
H(0) q)(0) = E(0) q)(0),
(4.5)
{H (2(z/T.); z)-- i 0~} 0 (z) = 0
(4.11)
contains two different time scales set by the laseror maser-period T = 2 n/co and the switching-on time
T,. Assuming that the amplitude 2(z/7;) changes negligibly during one period of the external field we treat
both time-dependencies separately:
First we determine the Floquet-eigenstates u~(2)
for each fixed 2-value in the range of the function
2 = 2(r/T~) as the solutions of the eigenvalue equation
Let the system be initially in an eigenstate of Ho
.,~ (,~;t)u~(,~; t)= ~(,~)u~(,t; t)
~,(T=0)=uj(;40)=0; ~=0)=~oj.
u~()~; t + T ) = u~(2; t),
(4.12)
where
Jr(2; t)= H ()~; t)-- it3r
After stationarity is reached for r > T,, because of the
completeness relation (2.12) the solution ¢'(z) of
Schr6dinger's equation (4.11) can be decomposed
(4.13)
O(z)=~aiui(2y; "c)e - i ~('~±)~,
This corresponds to integrating out the fast, oscillating time-dependence which is denoted by t in (4.12).
Secondly we include the slow, parametric timedependence of 2 =2(~/T~) which is determined by the
adiabatic evolution equation
~ (2 (z/T,); t) O(z; t)= i ~ ¢(z; t).
(4.14)
From ~(-c; t) the solution @(z) of the original Schrrdinger equation (4.11) can easily be recovered by
equating t and z
~b(v)= ~(v; t) lt=~,
(4.15)
since
= (]g ~) (z; "c)+ i(OtO) (z; z)
(4.16)
The main advantage of (4.14) lies in the fact that it
is a Schr6dinger equation precisely of the form of
(4.2), Floquet-eigenstates u~ play the role of the stationary eigenstates (p. Therefore, the conventional
adiabatic theorem can be applied: Assuming the initial value
(4.17)
the solution of (4.14) converges for T~~ ov to
v
• (~; t) = e-i oId,' ~o().)u~(),(z/T~); t).
(4.18)
Equating t and z now proves the adiabatic theorem
of the Floquet picture.
The same method of separating fast (t) and slow
(z) time-dependence has to be employed to calculate
deviations from adiabaticity. To be specific, let us consider an adiabatic parameter-variation of the form
//
0~
2(z) = { smoothly monotonically increasing,
2y = const.,
(4.21)
with z-independent amplitudes a~.
Calculations of the transition probabilities
P.i -~ = Iai Ie = I (ui (2y; t) l @(r)) 12
(4.22)
enables one to measure deviations from the full adiabatic behaviour characterised by [a~ !z -= ~;u- The solution of the evolution equation (4.14) corresponding
to ~ (z) is now given by
O(z; t)=~aiui(2y; Oe -i~'(~O~,
z > Ta,
(4.23)
i
2 1 T
tail =-~ °~ d t l(ui(2y; t)tO(r; t))12,
=HO(~;z)
O(z; t)[~= o = u~(2(0); t),
z> T~,
i
and hence the z-independent probabilities (4.22) may
be written
i ~ tfi (z) = i(O~O) (z; z) + i(~ t O) (Z ; T)
=H@(z).
(4.20)
~<0
0<z_<T,
I
T2>T
a .
(4.19)
(4.24)
and from Parseval's theorem we obtain the identity
[ai[2=~l((u~i,m)().~y;t)tcb(~;t)))l z,
v> T~.
(4.25)
m
Each term in this sum may be interpreted as a transition probability in the extended Hilbert space (cf.
Chap, 2); summing over all members of the same class
(cf. (2.9)) of a Floquet state yields the physical transition probability into this class.
It is (4.25) which contains the extended scalar
product that is appropriate for generalizing well
known formulae from standard theory [20] to the
Floquet picture. For example, to first order in 1/Ta
(not in 2 ]) we obtain
dz'~[('c') u(i,,,)(2 )
[
m
uj(2
-
0
z " e(i.m)(2)- ~j
exp
.
(4.26)
0
As we have shown in previous work [8, 9] deviations
from the adiabatic approximation mainly occur at
avoided crossings of the Floquet-eigenvalues e~
=~:~(2) and can be understood in close analogy to
the well-known Landau-Zener mechanism.
10
Let us therefore assume that there is an isolated
avoided crossing at 2~ < 2f. Since
E
(4.27)
is a slowly varying function of z, the main contribution to Jail2 arise from those summands in (4.26) for
which the oscillations of the exponential are minimal.
In particular, at the avoided crossing there is one
term, m0 say, for which both the frequency (~(i,mo)-ej)
is smallest and the overlap (4.27) largest; this term
suffices to saturate the sum in (4.26). Hence we see
that in this case the transition probability is determined by the energy gap within one Brillouin-zone
at an avoided crossing
68,=Jei+moco-ejl=min,,~zlei+mog-ejJ.
(4.28)
It is important to note that 6e may be substantially
smaller than the difference between those representatives that are continuously connected to the eigenvalues of Ho. Therefore the 'excitation energy' measured as the difference between static eigenvalues becomes irrelevant as has been discussed in detail in
E9-1.
A quantitative description of the dynamics at an
avoided crossing can be obtained from a two Floquetstate approximation. Again separating fast and slow
time-dependence, the usual Landau-Zener calculations can be carried out and yield for the transition
probability at an avoided crossing
P~~ i = e -g~
(4.29)
where the dimensionless parameter ~/is given by
q
6e62
=~--- ~
,
(4.30)
and 62 denotes the 2-interval in which the distance
between the levels is enlarged by a factor of If2 (see
Fig. 6). As could be expected the transition probability Pj_.~ is determined by a combination of spectral
properties 6 e. 6 2 and the velocity )~; again the appearance of 6 e, i.e. the spacing of Floquet indices within
one Brillouin-zone, is decisive for a Landau-Zener
transition among Floquet states [8, 9].
Let us now turn to the problem of highly excited
hydrogen atoms entering a microwave cavity. Obviously, the time scale T, necessary to ensure adiabaticity is of central importance. To estimate the order
of magnitude of T~, we again use the SSE model and
model the slowly rising field 'seen' by the atom by
~X
Fig. 6. Geometricalmeaningof the parameters 6e, 62 characterizing
the 8-2 hyperbolae
a gaussian
2=Z(z)= 2Iexpt2 s,
,
~<0,
z > 0.
(4.31)
We then assume that for -c- . - ov the stationary state
with principal quantum number no is populated and
solve the Schr6dinger equation (4.11); for z =0, when
the turn-on is completed, we perform a Floquet decomposition according to (4.21).
For a first application of this procedure we choose
co/2rc=9.92 GHz, n0=64 and 2 s = 1 5 V/cm. This is
an example of an 'uncritical" case: From Fig. 5 we
see that 864 does not undergo an avoided crossing
for 0<2_<2 s. However, the level no =64 exhibits a
substantial AC-Stark shift (in other terminology, the
corresponding state is strongly 'dressed') and hence
the Floquet state u64 at 2 s is considerably different
from the initial stationary eigenstate ~P64- The result
of the above described procedure for estimating T,
is shown in Fig. 7a where we have plotted la6412, the
occupation probability of the Floquet state continuously connected to the initial stationary state, as
a function of T,.
We see that a switching-on time of only a few
cycles results in almost complete adiabaticity and an
almost pure Floquet state propagates for z > 0 . Since
even the very fast hydrogen atoms used in the experiments [-2i] see the fringe field for some 40 to 80 periods, adiabaticity can be taken for granted. Therefore,
any model which neglects the effects of the fringe fields
leads to definitively wrong results: A sudden turn-on
of the field described by a 0-function determines the
11
[(~74 [2
I ~ 6 4 [2
I * l ~ l
I
~,
1.0
i
]
*
1.11
I~74 ]2
I
'
I
I ~ l ~ l
l
.t/
05
l ~ l ' l ~ I - -
0.5
I
1
a
1
2
]
4
I
5
0.r¢
O
b
I
I
*
3
~
5
H
,o
2o
3o
~o
5~
To/T
Fig. 7, a Occupation probability of the Ftoquet state continuously connected to the stationary initial state no=64 as a function of the
switching-on time T~ (cf. (4.31), T~=0 means a sudden, 0-function like turn-on) in units of the maser period T; 2i=15V/cm. b as a,
but no=74 and 2~=7 V/cm. e Occupation probability of the state marked by an arrow in Fig. 3b which seems to continue the initial
state no= 74 after several avoidedcrossings; 2s = 9.95 V/cm
amplitudes a~ in the decomposition (4.21) as the projections of the initial state on the final Floquet states
at the moment of switching-on and hence the resulting wavefunction is a superposition of many Floquet
states. The fact that this is not just a small error can
be clearly seen from Fig. 7 a: Whereas the initial state
stays in its continuously connected Floquet state with
probability la6412 >0.995 for T~= 2 T, a sudden turnon yields [a6412 ~0.39.
We stress that the simple adiabatic behaviour
shown in Fig. 7a becomes visible only as a result of
the Floquet decomposition (4.21) and is completely
obscured when decomposing with respect to stationary eigenstates of Ho.
However, the dynamics becomes more complicated when the field strength 21 is enhanced: Above a
critical value 2c(no), the quasienergy originating from
the stationary state no starts to exhibit many avoided
crossings. F o r example, from Fig. 3b we read off
2~(74) g 8 V/cm. Obviously, now many Landau-Zener
transitions occur when, during the entrance into the
cavity, 2(~) exceeds At(no).
To illustrate this phenomenon, we perform calculations analogous to those leading to Fig. 7a for n o
= 74 and 2f = 7 V/cm < 2c(74) (see Fig. 7b). Again, the
steep rise of [a74J 2 demonstrates the importance of
adiabaticity; this time, however, convergence to 1.0
is not as rapid as in Fig. 7a since the final state already 'feels' the influence of other Floquet states close
by in the Brillouin zone.
Next, we repeat this calculation using the same
initial state n0--74 but choose for the final field
strength 2 s = 9.95 V/cm > 2c(74) and plot the occupation probability of that state that seems to continue
the initial one after several avoided crossings (marked
by the arrow in Fig. 3b). In fact, this state has to
be denoted by the principal quantum number n--74
since, after an avoided crossing, the Floquet states
behave as if they had ignored the latter [22]. In
Fig. 7 c we observe a strong depopulation of that level: At each avoided crossing the jumping-probability
(4.29) is diminished with increasing T~. Therefore at
each avoided crossing the wavefunction tends to stay
on the continuously connected branch of the e - 2
hyperbola, it has time to change and to take on the
structure of the other Floquet state engaged in the
avoided crossing. Thus, it Jollows that adiabatic processes are responsible for transitions to higher states.
5. C o n c l u s i o n s
The experimentally observed microwave ionization of
highly excited hydrogen atoms poses serious problems to perturbative quantum mechanics: Since a
conventional perturbative expansion in the field
strength )0 would require the inclusion of about 100
orders, a non-perturbative ansatz is needed.
A viable scheme is provided by the Floquet picture since it incorporates the external field in a n o n -
12
perturbative manner and furthermore allows for a
clear description of adiabatic evolution. Any discussion of adiabatic dynamics has to start from an investigation of the spectral properties of a whole oneparameter family of quasi-hamiltonians
~t~(2),
40
'D
0<2<2z.
As our calculations show, up to certain critical values
of the field strength adiabaticity completely determines the solution of Schr6dinger's equation and
holds down to time scales which are even shorter
than those relevant for the experiments.
For a discussion of the ionization mechanism a
restriction to the space spanned by the localized
bound states seems to be insufficient, since there are
no normalizable Floquet states. It is, however, possible to include continuum effects in a qualitative manner since it is plausible that the imaginary part of
the complex Floquet indices describing the actual
Floquet-resonances is larger for resonances originating from higher bound states of Ho. Hence we now
regard our numerically calculated quasienergies for
the SSE model as an approximation to the real part
of the indices describing resonances and have to keep
in mind an additional imaginary part determining
their life-time.
Thus, the following picture of the microwave ionization of highly excited atoms finally emerges: A hydrogen atom prepared in a Rydberg state (with main
quantum number no, say) which enters a microwave
cavity sees a field which rises sufficiently slowly to
guarantee adiabaticity to a very high degree. As long
as the instantaneous field strength 2(0 stays below
a critical value 2~(no) which marks the onset of multiple avoided crossings, the initial state is adiabatically
shifted into the continuously connected Floquet-resonance:
~O,o(x)e- ie, ot-~ U,o(X , t) e-i~,,o '.
(s.1)
If the final field strength within the cavity is less then
2c(no), nothing happens: When the atom leaves the
cavity, it is adiabatically shifted back into the initial
state. The situation changes abruptly when, during
the entrance into the cavity, the field strength 2(0
reaches 2c(no): As described in Chap. 4, LandauZener type transitions among Floquet states take
place. It is a characteristic feature of the SSE spectra
that the first avoided crossings are not isolated but
are followed in a rapid succession by further ones.
Hence, if the final field strength 2 s is larger than 2~(n0),
the atom undergoes many Landau-Zener transitions
already when entering the cavity. In this way Floquet
6O
fi5
70
75
prine[p0[ qu0ntumnumber
Fig. 8. Stars: Calculatedcriticalfieldstrength2c(n)for the SSE model marking the onset of multiple avoided crossings (co/
2~=9.92 GHz). Boxes: Experimentallyfound 10%-ionizationfield
strength (no static field superimposed);data taken from Fig. 2 in
[213
states are populated which originate from higher lying
stationary states and therefore have quasienergies
with larger imaginary parts. Thus, as soon as these
Landau-Zener transitions set in the Schr6dinger
wavefunction ~9 becomes fast delocalizing, i.e. the
atom ionizes.
It should be emphasized that those transitions are
non-perturbative and their probabilities are determined by the Brillouin-zone distance ~e which is only
a small fraction of the frequency co. In contrast, timedependent perturbation theory would demand the absorption of a huge number of photons to account
for the ionization energy AE.
In order to treat the above discussed effects of
the continuum in a mathematically consistent way
one could employ the method of complex scaling introduced into the Floquet theory in [23].
However, there are implications of the above suggested ionization mechanism which can be checked
even within our simple model: The occurence of the
first avoided crossings must be related to the ionization threshold. Indeed, plotting the values 2c=2c(n)
from our calculations we obtain Fig. 8 which shows
that these critical values are remarkably close to the
experimentally found 10%-ionization field strength
[21]. Besides being a strong argument in favour of
our Landau-Zener mechanism this fact a posteriori
justifies the ansatz of concentrating on the bound
state dynamics.
We stress that these Landau-Zener transitions occur during the rise of the external field which therefore
13
is essential for the preparation of the state that actually ends up in the cavity. Hence, to model the entrance
into the microwave cavity by a time-dependence of
the form
(5.2)
)¢(t) = •f 0(t) sin(co t + 6)
is too crude an approximation: First of all, (5.2) implies that an initial state ~00 propagates as
O(x,O=~a~ui(x,t+ ~6\) e
_~,
i,
t>0,
(5.3)
where the amplitudes
(5.4)
contain a rather unphysical dependence on the phase
6 which expresses the fact that they are solely determined by the moment t = 0 of switching-on. Needless
to say, this artificial problem does not" occur in an
adiabatic approach.
Secondly, we have shown that excitation of higher
Floquet states results from adiabatic evolution at
avoided crossings. This important feature is completely ignored in approaches based on (5.2).
Notwithstanding these facts, the 'width function'
approach of Blfimet and Smilansky [7] has been
shown to yield values for the ionization threshold
which are in agreement with experimental data although the fringe fields are not taken into account.
This is quite easy to understand: The width function
measures the extent of Floquet states taken at fixed
time t = 0 (assuming an external field 2 sin cot). Obviously, the appearance of many avoided crossings
is reflected in the structure of the instantaneous Floquet states and, hence, in the width function. Thus,
critical field strengths can be determined from the
width function as well as from the quasienergy spectrum. However, the underlying physical mechanism
that we suggest is very different from that of Blfimel
and Smilansky: From their point of view, ionization
should occur after population of high lying eigenstates of H 0 ('window states') and a Floquet state
corresponding to a fixed field strength 21 should serve
to connect both the initial and the window states;
the fringe fields play no rble at all. In contrast, from
our calculations we have to conclude that an initial
state is adiabatically turned into a Floquet state when
entering the fringe fields, at avoided crossings fast decaying Floquet states are populated.
The detailed investigation of the dynamics of highly excited hydrogen atoms in the fringe fields of a
microwave cavity yields new physical insights, in par-
ticular, we expect measurable effects: Since the Landau-Zener 'jumping' probabilities (4.29) are affected
by the precise form of the fringe fields, different fi'inge
fields may lead to different occupation probabilities
of the final Floquet states in the cavity and hence
to different structures in the ionization signal. It will
be very interesting to study the results of thture experiments, where the fringe fields will be under control
[24], from this point of view.
It is worthwhile to note that the simple expression
(5.1) is essentially the quantum mechanical formulation of the 'stabilization mechanism' conjectured by
Jensen [25] on the basis of classical calculations.
Since this formula (5.1) is based on the adiabatic theorem of quantum mechanics it is valid for any value
of the scaled frequency no3co;however, the time scales
necessary for full adiabatic behaviour are longer in
near-resonant cases than in non-resonant ones.
Finally we should like to compare the quantum
mechanical SSE model with its classical counterpart.
It is well known that the classical SSE model exhibits
chaotic behaviour above certain values of the coupling strength which furthermore coincide astonishingly well with the observed ionization threshold [21].
From our quantum calculations we conclude that this
same threshold corresponds to a change in the quasienergy spectra from regular to irregular, i.e. from
smooth 2-dependence to an irregular avoided crossing structure. Hence we see that there is a deep connection between the character of the classical trajectories and the structure of the quantum mechanical
quasienergy spectra.
Such a connection has often been found for timeindependent hamiltonians and in particular been verified for the case of a hydrogen atom in a static magnetic field [26]; our investigations indicate that it is
also true in the Floquet picture for periodically timedependent hamiltonians. (For a discussion of this subject, see also [27].)
Again, it is instructive to study the harmonic oscillator in this context: Of course the classical version
is integrable and the Floquet spectrum (2.16) exhibits
no avoided crossings.
In order to understand the role of the classical
orbits we construct the classical analogue of the quasi-hamiltonian ~ = H ( t ) - i ~ t . As we have seen, within the Floquet picture - i ~ t has to be regarded as
the momentum Pt conjugated to the time coordinate
t. Hence we write for the classical quasi-hamiltonian:
Jd~cl=H(t)+pt.
(5.5)
Since periodicity is imposed on the time t, the dimensionless quantity cot must be interpreted geometrically as angle coordinate with period 2re. Thus the
14
phase space of the system is the cotangent bundle
of an infinite cylinder.
After transforming (5.5) to action-angle coordinates the usual EBK quantisation rules [28] can be
applied to yield the semi-classical Floquet indices.
As an example for this procedure we consider the
classical counterpart of the driven oscillator
~=l--
p2x+ p, + @
x2 + 2x sin cot.
P
(5.6)
Performing a canonical transformation to action-angle-variables
(Px, x; Pt, t) -~ (Io, (Po; I, g0)
(5.7)
we obtain
22
1
H'~l=c°°I°+°°I4 4~t ¢o2--co 2"
(5.8)
Quantising the libration
lo=n+¼et
with M a s l o v - i n d e x a~ = 2 a n d the r o t a t i o n
I = rn + ¼o~
with M a s l o v - i n d e x a t - - 0 we i m m e d i a t e l y recover the
exact q u a n t u m m e c h a n i c a l f o r m u l a (2.16).
This m e t h o d m a y serve as a viable p r o c e d u r e for
investigating the semi-classical limit of periodically
d r i v e n q u a n t u m systems. It s h o u l d be clear t h a t anh a r m o n i c i t i e s in the p o t e n t i a l of Ho p r o d u c e formidable difficulties, i.e. chaos in the classical case a n d
avoided crossings o r i g i n a t i n g from a r e s o n a n c e p o i n t
as in Fig. 4 i n the q u a n t u m case.
As a c o n c l u d i n g r e m a r k we stress the generality
of o u r results. A l t h o u g h we use the m o d e l of R y d b e r g
a t o m s i n s t r o n g m i c r o w a v e fields to s u p p o r t o u r considerations they are also relevant for other periodically driven q u a n t u m systems, for i n s t a n c e m a t t e r interacting with s t r o n g laser fields.
We should like to thank Prof. K. Dietz for initiating this work and
constructive criticism on the manuscript.
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